Simplify; express your answer in exponential form. Assume $p\neq 0, y\neq 0$. $\dfrac{{(p^{5}y^{-5})^{-3}}}{{(p^{3}y^{-5})^{4}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(p^{5}y^{-5})^{-3} = (p^{5})^{-3}(y^{-5})^{-3}}$ On the left, we have ${p^{5}}$ to the exponent ${-3}$ . Now ${5 \times -3 = -15}$ , so ${(p^{5})^{-3} = p^{-15}}$ Apply the ideas above to simplify the equation. $\dfrac{{(p^{5}y^{-5})^{-3}}}{{(p^{3}y^{-5})^{4}}} = \dfrac{{p^{-15}y^{15}}}{{p^{12}y^{-20}}}$ Break up the equation by variable and simplify. $\dfrac{{p^{-15}y^{15}}}{{p^{12}y^{-20}}} = \dfrac{{p^{-15}}}{{p^{12}}} \cdot \dfrac{{y^{15}}}{{y^{-20}}} = p^{{-15} - {12}} \cdot y^{{15} - {(-20)}} = p^{-27}y^{35}$